Chapter 8: Q2E (page 433)

In problems 1-6, determine the convergence set of the given power series.
$\sum_{n = 0}^{\infty} \frac{3^{n}}{n!} x^{n}$

Short Answer

Expert verified

The set is, $x \in (- \infty, \infty)$ .

Step by step solution

Step 1:To Find the Radius of convergence

Using the ratio test to determine the radius of convergence.

$\begin{matrix} \lim_{n \to \infty} | \frac{a_{n}}{a_{n + 1}} | = \lim_{n \to \infty} | \frac{\frac{3^{n}}{n!}}{\frac{3^{n + 1}}{n + 1!}} | \\ = \lim_{n \to \infty} | \frac{3^{n}}{3^{n + 1}} \times \frac{(n + 1)!}{n!} | \\ = \lim_{n \to \infty} | \frac{(n + 1) n!}{n!} \times \frac{3^{n}}{3 \times 3^{n}} | \\ = \lim_{n \to \infty} | \frac{n + 1}{3} | \\ = \infty \end{matrix}$

The radius of convergence is $\infty$ , therefore the series is convergent over the complete real line.

$| x | < \infty$

Step 2:Find the set of convergence

The convergent set for the given power series is

$x \in (- \infty, \infty)$

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In problems 1-6, determine the convergence set of the given power series.
$\sum_{n = 0}^{\infty} \frac{3^{n}}{n!} x^{n}$

Short Answer

Step by step solution

Step 1:To Find the Radius of convergence

Step 2:Find the set of convergence

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In problems 1-6, determine the convergence set of the given power series.∑n=0∞3nn!xn

Short Answer

Step by step solution

Step 1:To Find the Radius of convergence

Step 2:Find the set of convergence

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

In problems 1-6, determine the convergence set of the given power series.
$\sum_{n = 0}^{\infty} \frac{3^{n}}{n!} x^{n}$